P-adic Sets Research Articles

Good functions, measures, and the Kleinbock–Tomanov conjecture

Abstract In this paper, we prove a conjecture of Kleinbock and Tomanov (2007) on Diophantine properties of a large class of fractal measures on Q p n \mathbb{Q}_{p}^{n} . More generally, we establish the 𝑝-adic analogues of the influential results of Kleinbock, Lindenstrauss and Weiss (2004) on Diophantine properties of friendly measures. We further prove the 𝑝-adic analogue of one of the main results of Kleinbock (2008) concerning Diophantine inheritance of affine subspaces, which answers a question of Kleinbock. One of the key ingredients in the proofs of Kleinbock, Lindenstrauss and Weiss is a result on ( C , α ) (C,\alpha) -good functions whose proof crucially uses the Mean Value Theorem. Our main technical innovation is an alternative approach to establishing that certain functions are ( C , α ) (C,\alpha) -good in the 𝑝-adic setting. We believe this result will be of independent interest.

Values of p-adic hypergeometric functions and p-adic analogue of Kummer’s linear identity

Let p be an odd prime and [Formula: see text] be the finite field with p elements. This paper focuses on the study of the values of a generic family of hypergeometric functions in the p-adic setting, which we denote by [Formula: see text], where [Formula: see text] and [Formula: see text]. These values are expressed in terms of numbers of zeros of certain polynomials over [Formula: see text]. These results lead to certain p-adic analogues of classical hypergeometric identities. Namely, we obtain p-adic analogues of particular cases of a Gauss’s theorem and a Kummer’s theorem. Moreover, we examine the zeros of these functions. For example, if n is odd, we characterize t for which [Formula: see text] has zeros. In contrast, we show that if n is even, then the function [Formula: see text] has no zeros for any prime p apart from the trivial case when [Formula: see text].

Bi-algebraicity in the rank one Riemann-Hilbert correspondence via o-minimality

For a smooth, projective, complex algebraic variety X, the Riemann–Hilbert correspondence establishes a complex analytic isomorphism between the ‘Betti moduli space’ of rank n local systems on Xan and the ‘de Rham moduli space’ of rank n vector bundles with flat connection on X. In the rank one case, C. Simpson precisely characterizes the subvarieties of these moduli spaces that are ‘bi-algebraic’ for this typically transcendental, analytic isomorphism. In this short note, we give a new proof of this characterization of Simpson, using methods from o-minimal geometry. We adapt the o-minimal proof to a p-adic setting, namely that of Mumford curves.

Exploring tropical differential equations

Abstract The purpose of this paper is fourfold. The first is to develop the theory of tropical differential algebraic geometry from scratch; the second is to present the tropical fundamental theorem for differential algebraic geometry, and show how it may be used to extract combinatorial information about the set of power series solutions to a given system of differential equations, both in the archimedean (complex analytic) and in the non-Archimedean (e.g., p-adic) setting. A third and subsidiary aim is to show how tropical differential algebraic geometry is a natural application of semiring theory, and in so doing, contribute to the valuative study of differential algebraic geometry. We use this formalism to extend the fundamental theorem of partial differential algebraic geometry to the differential fraction field of the ring of formal power series in arbitrarily (finitely many variables; in doing so we produce new examples of non-Krull valuations that merit further study in their own right.

Simple modules for quiver Hecke algebras and the Robinson–Schensted–Knuth correspondence

We formalize some known categorical equivalences to give a rigorous treatment of smooth representations of p-adic general linear groups, as ungraded modules over quiver Hecke algebras of type A. Graded variants of RSK-standard modules are constructed for quiver Hecke algebras. Exporting recent results from the p-adic setting, we describe an effective method for construction and classification of all simple modules as quotients of modules induced from maximal homogenous data. It is established that the products involved in the RSK construction fit the Kashiwara-Kim notion of normal sequences of real modules. We deduce that RSK-standard modules have simple heads, devise a formula for the shift of grading between RSK-standard and simple self-dual modules, and establish properties of their decomposition matrix, thus confirming expectations for p-adic groups raised in a work of the author with Lapid. Subsequent work will exhibit how the presently introduced RSK construction generalizes the much-studied Specht construction, when inflated from cyclotomic quotient algebras.

Sato-Tate distribution of p-adic hypergeometric functions

Recently Ono, Saad and the second author [21] initiated a study of value distribution of certain families of Gaussian hypergeometric functions over large finite fields. They investigated two families of Gaussian hypergeometric functions and showed that they satisfy semicircular and Batman distributions. Motivated by their results we aim to study distributions of certain families of hypergeometric functions in the p-adic setting over large finite fields. In particular, we consider two and six parameters families of hypergeometric functions in the p-adic setting and obtain that their limiting distributions are semicircular over large finite fields. In the process of doing this we also express the traces of pth Hecke operators acting on the spaces of cusp forms of even weight kge 4 and levels 4 and 8 in terms of p-adic hypergeometric function which is of independent interest. These results can be viewed as p-adic analogous of some trace formulas of [1, 2, 6].

Zeros of hypergeometric functions in the p-adic setting

McCarthy (Pac J Math 261(1):219–236, 2013) defined hypergeometric functions in the p-adic setting over finite fields using p-adic gamma functions. These functions possess many properties that are analogous to classical hypergeometric type identities. In this paper, we investigate values of two generic families of these hypergeometric functions that we denote by {_nG_n}(t)_p and {_n{widetilde{G}}_n}(t)_p for nge 3, and tin {mathbb {F}}_p, the finite field with p elements. These results generalize special cases of p-adic analogues of Whipple’s theorem and Dixon’s theorem of classical hypergeometric series. We also examine zeros of the functions {_nG_n}(t)_p, and {_n{widetilde{G}}_n}(t)_p over {mathbb {F}}_p. Moreover, we classify the values of t for which {_nG_n}(t)_p=0 for infinitely many primes. For example, we show that there are infinitely many primes for which {_{2k}G_{2k}}(-1)_p=0. In contrast, for tne 0 there are no primes for which {_{2k}{widetilde{G}}_{2k}}(t)_p=0.

P-adic quotient sets: diagonal forms

For a set of integers A, we consider \(R(A)=\{a/b: a, b\in A, b\ne 0\}\). It is an open problem to study the denseness of R(A) in the p-adic numbers when A is the set of nonzero values attained by an integral form. This problem has been answered for quadratic forms. Very recently, Antony and Barman have studied this problem for the diagonal binary cubic forms \(ax^3+by^3\), where a and b are integers. In this article, we study this problem for diagonal forms. We extend their results to the diagonal binary forms \(ax^n+by^n\) for all \(n\ge 3\). We also study p-adic denseness of quotients of nonzero values attained by diagonal forms of degree \(n\ge 3\), where \(\gcd (n,p(p-1))=1\).

P-adic quotient sets: cubic forms

For \(A\subseteq \{1, 2, \ldots \}\), we consider \(R(A)=\{a/b: a, b\in A\}\). It is an open problem to study the denseness of R(A) in the p-adic numbers when A is the set of nonzero values assumed by a cubic form. We study this problem for the cubic forms \(ax^3+by^3\), where a and b are integers. We also prove that if A is the set of nonzero values assumed by a non-degenerate, integral, and primitive cubic form with more than 9 variables, then R(A) is dense in \({\mathbb {Q}}_p\).

On the wavefront sets associated with theta representations

We study a conjectural formula for the maximal elements in the wavefront set associated with a theta representation of a covering group over p-adic fields. In particular, it is shown that the formula agrees with the existing work in the literature for various families of groups. We also recapitulate the results of an analogous formula in the archimedean case, which motivated the conjectural formula in the p-adic setting.

Quantum invariants for decomposition problems in type A rings of representations

We prove a combinatorial rule for a complete decomposition, in terms of Langlands parameters, for representations of p-adic GLn that appear as parabolic induction from a large family (ladder representations). Our rule obviates the need for computation of Kazhdan-Lusztig polynomials in these cases, and settles a conjecture posed by Lapid.These results are transferrable into various type A frameworks, such as the decomposition of convolution products of homogeneous KLR-algebra modules, or tensor products of snake modules over quantum affine algebras.The method of proof applies a quantization of the problem into a question on Lusztig's dual canonical basis and its embedding into a quantum shuffle algebra, while computing numeric invariants which are new to the p-adic setting.

P-Adic Multiwavelet Sets

This paper contains a brief review of p-adic MRA theory along with the introduction of p-adic multiwavelet set. Here we have studied various properties of p-adic multiwavelet sets such as disjointness of their dilates, counting formula for the elements in a wavelet set etc. and proved some results analogous to real setting, in special cases.

P-Adic Brownian Motion as a Limit of Discrete Time Random Walks

The p-adic diffusion equation is a pseudo differential equation that is formally analogous to the real diffusion equation. The fundamental solutions to pseudo differential equations that generalize the p-adic diffusion equation give rise to p-adic Brownian motions. We show that these stochastic processes are similar to real Brownian motion in that they arise as limits of discrete time random walks on grids. While similar to those in the real case, the random walks in the p-adic setting are necessarily non-local. The study of discrete time random walks that converge to Brownian motion provides intuition about Brownian motion that is important in applications and such intuition is now available in a non-Archimedean setting.

P-adic quotient sets II: Quadratic forms

For A⊆{1,2,…}, we consider R(A)={a/a′:a,a′∈A}. If A is the set of nonzero values assumed by a quadratic form, when is R(A) dense in the p-adic numbers? We show that for a binary quadratic form Q, R(A) is dense in Qp if and only if the discriminant of Q is a nonzero square in Qp, and for a quadratic form in at least three variables, R(A) is always dense in Qp. This answers a question posed by several authors in 2017.

A formula for the geometric Jacquet functor and its character sheaf analogue

Let (G,K) be a symmetric pair over the complex numbers, and let X=K\G be the corresponding symmetric space. In this paper we study a nearby cycles functor associated to a degeneration of X to MN\G, which we call the "wonderful degeneration". We show that on the category of character sheaves on X, this functor is isomorphic to a composition of two averaging functors (a parallel result, on the level of functions in the p-adic setting, was obtained in [BK, SV]). As an application, we obtain a formula for the geometric Jacquet functor of [ENV] and use this formula to give a geometric proof of the celebrated Casselman's submodule theorem and establish a second adjointness theorem for Harish-Chandra modules.

Quotients of Fibonacci Numbers

There have been many articles in the MONTHLY on quotient sets over the years. We take a first step here into the p-adic setting, which we hope will spur further research. We show that the set of quotients of nonzero Fibonacci numbers is dense in the p-adic numbers for every prime p.

Summation identities and special values of hypergeometric series in the p-adic setting

We prove hypergeometric type summation identities for a function defined in terms of quotients of the p-adic gamma function by counting points on certain families of hyperelliptic curves over Fq. We also find certain special values of that function.

Certain transformations for hypergeometric series in the p-adic setting

In [The trace of Frobenius of elliptic curves and the p-adic gamma function, Pacific J. Math. 261(1) (2013) 219–236], McCarthy defined a function nGn[⋯] using the Teichmüller character of finite fields and quotients of the p-adic gamma function. This function extends hypergeometric functions over finite fields to the p-adic setting. In this paper, we give certain transformation formulas for the function nGn[⋯] which are not implied from the analogous hypergeometric functions over finite fields.

On P-Adic λ-Model on the Cayley Tree II: Phase Transitions

In the present paper, we consider the p-adic λ-model on the Cayley tree of order two. It is considered generalized p-adic quasi Gibbs measure depending on a parameter ρ ∈ ℚp, for the λ-model. In the p-adic setting there are several kinds of phase transitions such as storing phase transition, phase transition in terms of the generalized p-adic quasi Gibbs measures. In the paper, we consider two regimes with respect to the parameter ρ. The existence of generalized p-adic Gibbs measures in both regimes is proved. We prove the existence of the phase transition for the p-adic λ model on the Cayley tree of order two in the first regime. It turns out that in the second regime, we are able to establish the strong phase transition for a class of λ-models on the same tree. To prove the main results, we employ the methods of p-adic analysis, and therefore, our results are not valid in the real setting.

On $$p$$ p -adic lattices and Grassmannians

It is well-known that the coset spaces G(k((z)))/G(k[[z]]), for a reductive group G over a field k, carry the geometric structure of an inductive limit of projective k-schemes. This k-ind-scheme is known as the affine Grassmannian for G. From the point of view of number theory it would be interesting to obtain an analogous geometric interpretation of quotients of the form G(W(k)[1/p])/G(W(k)), where p is a rational prime, W denotes the ring scheme of p-typical Witt vectors, k is a perfect field of characteristic p and G is a reductive group scheme over W(k). The present paper is an attempt to describe which constructions carry over from the function field case to the p-adic case, more precisely to the situation of the p-adic affine Grassmannian for the special linear group G=SL_n. We start with a description of the R-valued points of the p-adic affine Grassmannian for SL_n in terms of lattices over W(R), where R is a perfect k-algebra. In order to obtain a link with geometry we further construct projective k-subvarieties of the multigraded Hilbert scheme which map equivariantly to the p-adic affine Grassmannian. The images of these morphisms play the role of Schubert varieties in the p-adic setting. Further, for any reduced k-algebra R these morphisms induce bijective maps between the sets of R-valued points of the respective open orbits in the multigraded Hilbert scheme and the corresponding Schubert cells of the p-adic affine Grassmannian for SL_n.